Complex and Nonlocal Systems

该提案涉及从复杂流体到社会网络等各种环境中产生的复杂和非局部系统。研究微观物体影响其宏观环境的混合系统是非线性科学的一个基本组成部分。该项目考虑非线性偏微分方程模型，其中微观部分由从底层随机方程导出的动力学方程和宏观流体方程来建模。该方案针对弱解、混合解、长时间行为解和统计解的奇点形成、存在唯一性或不存在性等基本问题。该提案还涉及主动网络的研究，自然或人工网络在响应网络依赖过程中进化。本文的主要目标是研究非局部相互作用和反馈对复杂系统演化的动力学影响。该方案解决了在复杂非局部系统的建模、计算和理论中出现的问题。这样的系统出现在各种各样的环境中，从复杂的物质，如血液或细胞内液体，到复杂的关系系统，如社会网络。复杂物质的特点是存在复杂的、相互作用的网络，这些网络由嵌入、影响并反过来影响基质或溶剂的非常小的物体组成。对这些相互作用的基本理解对于诸如制造具有极端特性的材料或在细胞水平上递送药物等应用的进展至关重要。具有反馈的信息通过这种网络进行扩散，对于研究基本生物过程和人工网络都具有重要意义。该建议的目的是揭示从局部和非局部相互作用中出现的系统规模质的变化的基本机制。

The proposal addresses complex and nonlocal systems, arising in a variety of contexts, ranging from complex fluids to social networks. The study of mixed systems in which microscopic objects influence their macroscopic environment is a fundamental component of nonlinear science. The project considers nonlinear PDE models in which the microscopic part is modelled by kinetic equations derived from underlying stochastic equations, coupled with macroscopic fluid equations. The proposal aims at basic questions of singularity formation, existence and uniqueness or lack thereof for weak solutions, mixing, long time behavior and statistical solutions. The proposal is concerned also with the study of active networks, natural or artificial networks that evolve in response to network-dependent processes. The overarching goal of the proposal is to study the dynamical effects of nonlocal interaction and feedback on evolving complex systems.The proposal addresses problems that arise in the modeling, computation and theory of complex nonlocal systems. Such systems occur in a great variety of circumstances, ranging from complex matter, such as blood or intracellular fluids, to complex relational systems, such as social networks. Complex matter is characterized by the presence of complex, interacting networks of very small objects being embedded, influenced, and in turn influencing a matrix or a solvent. The basic understanding of these interactions is crucial for progress in applications such as the manufacturing of materials with extreme properties or the delivery of drugs at the cellular level. Diffusion of information with feedback through such networks is of great importance for the study of basic biological processes and also for the study of artificial networks. The aim of the proposal is to uncover fundamental mechanisms of system-size qualitative changes emerging from local and nonlocal interactions.